Blower motor quick working while driving down the highway on my 2006 Chevy Avalanche. I checked all fuses and relays. All Ok. Could this be a resister? or is it a blower moter gone bad?

limited info generic answer the cat is usually the last and driving 2 hrs is not going to do it and usually need to do quick drive cycle 2 or 3 times -----or over 125 miles of alternate driving --https://smogcheckvacaville.com/general-motors-drive-cycle-for-smog-check/ ----------- https://repairpal.com/how-to-perform-a-basic-drive-cycle ---------

I looked at 05 avalanche 1500--automatic hvac. The blower fuse is hot all the time. The fuse has an input leg and output leg, even if the fuse is blown, the input leg would still have voltage. If no voltage, you could use jumper wire connected to voltage on the output side of fuse cavity to see if blower would work. If you use jumper, use an inline fuse.
Yes, it could be something wrong with fuse box or wiring circuit?

If a/c does not blow, check the blower fuse and high speed relay. If all checks good, you may have a bad blower motor. Sometimes, hitting on top of the passenger side dash top, the blower motor may come on. If that works, you still will need to replace blower motor.
If a/c doesn't blow cold but the blower motor runs......... check the front a/c condenser located in front of radiator for a hole that was caused by a rock from the highway that could have punctured it. That's a common problem.

blower motor resister located just below blower fan on passenger side . it is hidden by plastic cover

your blower motor has gone bad it will need to be replaced under the dash on the passenger side remove the cover under the dash to access blower motor should be four screws and one plug to remove

It almost sounds like you have two issues going on. Ok. While you are driving you have heat and then when you stop it gets cold. That sounds like either air in system or a bad thermostat. And second issue is if you are not hearing the blower motor then you need to figure out whether you are getting power to blower motor and resistor. If you are getting power replace blower motor and since you are down there go ahead and replace resistor. If you are not getting power then you need to check for blown fuses and relays.

from pictures I'm seeing, it looks like it's next to the blower. Remove the insulating panel above the passenger's feet and look up. See if it looks like the picture below:

you may need a blower resistor and motor, the motors go bad and draw to many amps which melt the connectors to the motor/ resistor. replace the motor and resistor/ harness.

check the fuses, your owners manual will show where they are located, if this isn't the problem then you may need to replace your blower motor, a simple job in most vehicles requiring a minimal amount of tools and a little electronics knowledge

Their is a relay/ fuse box under the hood thir is a relay that is probbly sticking corrison on the contacts inside is a commen problem remove the relay and try switching it with a twin relay that controls somethig else in the box and see if it shuts off the fan if so replace the relay avalible at most auto part stores.